Commutativity preserving linear maps on central simple algebras

In [3] some constructions of exceptional Jordan algebras due to H. Freudenthal, T. A. Springer, and J. Tits were carried over to quadratic Jordan algebras (as in [4]) of arbitrary characteristic. The question was left open whether the Tits Constructions yielded all exceptional finite-dimensional central simple algebras in characteristic 2 (it was known that they do for characteristics #2). In this paper we settle this question in the affirmative. This result completes the structure theory for finite-dimensional quadratic Jordan algebras. The Tits Constructions as given in [3] involve the construction of a norm form. To prove that all the exceptional algebras arise from these constructions we need to show that all such algebras have a suitable norm form. This necessitates a slight detour in ??1 and 2 to verify that Jordan algebras in characteristic 2 have generic norms with the same properties as in the other characteristics. In ?3 we define the centroid for quadratic Jordan algebras and the corresponding notion of central simple algebras. In the next section we establish certain conditions under which a central simple Jordan algebra remains simple upon extension of the base field. In ?5 we apply these results to show that every exceptional finite-dimensional central simple algebra is a form of the 27-dimensional exceptional algebra ?((i3) (i.e. becomes ( upon suitable extension of the base field). In the final section we show that the Tits Constructions yield precisely all the exceptional finite-dimensional central simple Jordan algebras. Our proofs will be valid for all characteristics.

Let A be a finite dimensional central simple algebra. By the Skolem–Noether theorem, every automorphism of A is inner. We will give a short proof of a somewhat more general result. The concept behind this proof is the fact that every linear map on A belongs to the multiplication algebra of A. As an application we will describe linear maps α, β : A → A such that α(x)β(y) = 0 whenever xy = 0.

Let k be a field, K/k be a quadratic separable field extension, and 𝒜 a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically closed residue field of characteristic zero and the degree of 𝒜 is odd, we prove that all K/k-involutions on 𝒜 are cyclic.

Generalized identities of finite dimensional central simple algebras

Letdbe a non-zero derivation on a primitive ringRandƒ(x1,…,xn) a homogeneous polynomial of degreem. We prove that the conditiond(ƒ(r1,…,rn)t) = 0, for allr1,…,rn∈R, withtdepending onr1,…,rn, forcesRto be a finite dimensional central simple algebra andƒpower-central valued onR. We also obtain bounds on [R:Z(R)] in terms ofm.

We show that every central simple algebra $A$ over a field $k$ is Brauer equivalent to a quotient of a finite dimensional Hopf algebra over the same field. This shows that the natural generalization of the Schur group for Hopf algebras (which we call the Hopf Schur group) is in fact the entire Brauer group of $k$. If the characteristic of the field is zero, or if the algebra has a Galois splitting field with certain properties, we can take this Hopf algebra to be semisimple. We also show that if $F$ is any finite separable extension of $k$, then $F$ is a quotient of a finite dimensional commutative semisimple and cosemisimple Hopf algebra over $k$.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

It is elementary that a primitive commutative ring is a field. Kaplansky’s Theorem replaces “commutative” with “satisfies a polynomial identity” and says that a primitive PI-ring is a finite-dimensional central simple algebra over a field.

The authors present a new way to construct finite dimensional estimation algebra of nonmaximal rank. For a given finite dimensional estimation algebra E, one can construct a finite dimensional estimation algebra E' of nonmaximal rank which is isomorphic to E. The interesting fact is that even if E comes from a linear filtering system, there are arbitrary nonlinear filtering systems associated to E. Since every finite dimensional estimation algebra gives rise to a finite dimensional filter, we have constructed quite a large class of finite dimensional nonlinear filters.

In 1993 I. P. Shestakov came up with the question whether there exists a central simple finite-dimensional algebra over a field of characteristic 0, whose identities are not given by a finite set (Dniester Notebook, Question 3.103). In 2012 I. M. Isaev and the author constructed a desired example, giving an affirmative answer to the question posed. Here research into Shestakov’s question is continued for the case of commutative algebras. A seven-dimensional central simple commutative algebra over a field of characteristics 0 having no finite basis of identities is exemplified.

R. W. Brockett (1983) proposed to classify all finite dimensional estimation algebras. An affirmative solution to Brockett's problem will allow construction of all possible finite dimensional recursive filters from the Lie algebraic point of view. The concept of an estimation algebra with maximal rank was introduced by L. F. Tam et al. (1990). This is the most important general subclass of estimation algebras. S. S.-T. Yau and W.L. Chiou (1991) have already classified all maximal rank finite dimensional estimation algebras with state space dimension at most 2. Here, the case for state space dimension 3 is studied. >

A Noether Skolem theorem for group-graded rings

It is proved that every finite-dimensional algebra is embeddable in a simple finite-dimensional algebra (a suitable isotope of a matrix algebra). An isotope of the 2nd order matrix algebra over an infinite extension of the ground field may contain a trivial ideal. Every one-sided isotope of a simple unital alternative or Jordan algebra is a simple algebra. Besides, any isotope of a central simple non-Lie Maltsev algebra of characteristic other than 2 and 3 is a simple algebra. But an isotope of a simple Jordan algebra of the symmetric bilinear form on the infinite dimensional space may contain a trivial ideal.

We study the notion of the Killing form for Frobenius Lie algebras of dimension . The Killing form is a symmetric bilinear form on a finite dimensional Lie algebra over a field defined by where is denoted the trace and is an adjoint representation of . A Lie algebras is said to be semisimple if it has the nondegenerate Killing form. The research aims to consider the criterion for semisimplicity of Frobenius Lie algebras of dimension by using the Killing form. The results show that each Frobenius Lie algebra of dimension and is not semisimple since the the Killing form is degenerate or in other words, a determinant of a representation matrix of the Killing form is equal to zero. For the future research, it is still an open problem to consider the general formulas of the Killing form for higher dimensional Frobenius Lie algebra whether degenerate or nondegenerate such that the semisimplicity of a Lie algebra can be considered. We conjecture that each finite dimensional real Frobenius Lie algebra is not semisimple.

On trace forms of higher degree